Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Three-Dimensional Elasticity Analysis of Thick Rectangular Laminated Composite Plates Using Meshless Local Petrov-Galerkin (MLPG) Method

Abstract: This article presents apparently the first application of Meshless local Petrov-Galerkin (MLPG) method for 3-D elasticity analysis of moderately thick rectangular laminated plates. As with other Meshless methods, the problem domain is represented by a set of spread nodes in all three dimensions of the plate without configuration of elements. The Moving Least-Squares (MLS) method is applied to construct the required shape functions. A local asymmetric weak formulation of the problem is developed and MLPG is applied to solve the governing equations. Direct interpolation method is employed to enforce essential boundary conditions. Details of formulation, numerical procedure, convergence and accuracy characteristics of the method are investigated. Results are compared, where possible, with other analytical and numerical methods and show good agreement.

Localized MQ-RBF meshless techniques for modeling unsaturated flow

Abstract: In this study, we focus on space-time mesh-free numerical techniques for efficiently solving the Richards equation which is often used to model unsaturated flow through porous media. We propose an efficient approach which combines the use of local multiquadric (MQ) radial basis function (RBF) methods and space-time techniques. The localized MQ-RBFs meshless methods allow to avoid mesh generation and ill-conditioning problem where a sparse matrix is obtained for the global system which has the advantage of using reduced memory and computational time. To further reduce the computational cost, we use the space-time techniques having the advantages of solving the resulting algebraic system only once and removing the time-integration procedure. The proposed method has the benefit of considering collocation points on the boundaries of computational domains which makes it more flexible in dealing with complex geometries. We implement the proposed numerical model of infiltration and we perform a series of numerical tests, encompassing various nontrivial solutions, to confirm the performance of the proposed techniques. The numerical simulations show the accuracy, efficiency in terms of computational cost, and capability of the proposed numerical techniques in solving the Richards equation in two-, three- and four-dimensional space-time domains with complex boundaries.

Free Mesh Method: fundamental conception, algorithms and accuracy study

Abstract: The finite element method (FEM) has been commonly employed in a variety of fields as a computer simulation method to solve such problems as solid, fluid, electro-magnetic phenomena and so on. However, creation of a quality mesh for the problem domain is a prerequisite when using FEM, which becomes a major part of the cost of a simulation. It is natural that the concept of meshless method has evolved. The free mesh method (FMM) is among the typical meshless methods intended for particle-like finite element analysis of problems that are difficult to handle using global mesh generation, especially on parallel processors. FMM is an efficient node-based finite element method that employs a local mesh generation technique and a node-by-node algorithm for the finite element calculations. In this paper, FMM and its variation are reviewed focusing on their fundamental conception, algorithms and accuracy.

Finite increment gradient stabilization of point integrated meshless methods for elliptic equations

Abstract: This paper describes a new technique to stabilize meshless methods used in conjunction with point-based integration. The method proposed is based on the finite increment calculus (FIC) concepts for convection-dominated problems. In this paper a finite increment gradient operator is defined in such a way that second-order derivatives are included. This operator is then used in the context of a variational formulation of an elliptic problem in order to define a stabilized numerical procedure. For simplicity, the Poisson equation will be used in this paper to illustrate the method, although more general elliptic problems can be equally treated. An eigenvalue analysis will be carried out in order to demonstrate that no mechanisms are present in the resulting equations. Finally, a simple example will illustrate the technique. Copyright © 2000 John Wiley & Sons, Ltd.

Extension of Meshless Galerkin/Petrov-Galerkin Approach without Using Lagrange Multipliers

Abstract: By directly discretizing the weak form used in the finite element method, meshless methods have been derived. Neither the Lagrange multiplier method nor the penalty method is employed in the derivation of the methods. The resulting methods are divided into two groups, depending on whether the discretization is based on the Galerkin or the Petrov-Galerkin approach. Each group is further subdivided into two groups, according to the method for imposing the essential boundary condition. Hence, four types of the meshless methods have been formulated. The accuracy of these methods is illustrated for two-dimensional Poisson problems.